McKean–Vlasov process

In probability theory, a McKean–Vlasov process is a stochastic process described by a stochastic differential equation where the coefficients of the diffusion depend on the distribution of the solution itself. The equations are a model for Vlasov equation and were first studied by Henry McKean in 1966. It is an example of propagation of chaos, in that it can be obtained as a limit of a mean-field system of interacting particles: as the number of particles tends to infinity, the interactions between any single particle and the rest of the pool will only depend on the particle itself.

Definition

Consider a measurable function $\sigma :{\mathbb{ℝ}}^d\times {𝒫}({\mathbb{ℝ}}^d)\to {{ℳ}}_d(\mathbb{ℝ})$ where ${𝒫}({\mathbb{ℝ}}^d)$ is the space of probability distributions on ${\mathbb{ℝ}}^d$ equipped with the Wasserstein metric $W_2$ and ${{ℳ}}_d(\mathbb{ℝ})$ is the space of square matrices of dimension $d$. Consider a measurable function $b:{\mathbb{ℝ}}^d\times {𝒫}({\mathbb{ℝ}}^d)\to {{ℳ}}_d(\mathbb{ℝ})$. Define $a(x,\mu ):=\sigma (x,\mu )\sigma (x,\mu {)}^T$.

A stochastic process $(X_t{)}_{t\ge 0}$ is a McKean–Vlasov process if it solves the following system:

• $X_0$ has law $f_0$
• $d{X}_t=a(X_t,{\mu }_t)d{B}_t+b(X_t,{\mu }_t)dt$

where ${\mu }_t={ℒ}(X_t)$ describes the law of $X$ and $dB$ denotes the Wiener process. This process is non-linear, in the sense that the dynamics of ${\mu }_t$ do not depend linearly on ${\mu }_t$.

Existence of a solution

The following Theorem can be found in.

Existence of a solution — Suppose $b$ and $\sigma$ are globally Lipschitz, that is, there exists a constant $C>0$ such that:

$|b(x,\mu )-b(y,\nu )|+|\sigma (x,\mu )-\sigma (y,\nu )|\le C(|x-y|+W_2(\mu ,\nu ))$

where $W_2$ is the Wasserstein metric.

Suppose $f_0$ has finite variance.

Then for any $T>0$ there is a unique strong solution to the McKean-Vlasov system of equations on $[0,T]$. Furthermore, its law is the unique solution to the non-linear Fokker–Planck equation:

${\partial }_t{\mu }_t(x)=-\mathrm{\nabla }\cdot \{b(x,{\mu }_t){\mu }_t\}+\frac{1}{2}\sum _{i,j=1}^d{\partial }_{x_i}{\partial }_{x_j}\{a_{ij}(x,{\mu }_t){\mu }_t\}$

Propagation of chaos

The McKean-Vlasov process is an example of propagation of chaos. What this means is that many McKean-Vlasov process can be obtained as the limit of discrete systems of stochastic differential equations $(X_t^i{)}_{1\le i\le N}$.

Formally, define $(X^i{)}_{1\le i\le N}$ to be the $d$-dimensional solutions to:

• $(X_0^i{)}_{1\le i\le N}$ are i.i.d with law $f_0$
• $d{X}_t^i=a(X_t^i,{\mu }_{X_t})d{B}_t^i+b(X_t^i,{\mu }_{X_t})dt$

where the $(B^i{)}_{1\le i\le N}$ are i.i.d Brownian motion, and ${\mu }_{X_t}$ is the empirical measure associated with $X_t$ defined by ${\mu }_{X_t}:=\frac{1}{N}\sum _{1\le i\le N}{\delta }_{X_t^i}$ where $\delta$ is the Dirac measure.

Propagation of chaos is the property that, as the number of particles $N\to +\mathrm{\infty }$, the interaction between any two particles vanishes, and the random empirical measure ${\mu }_{X_t}$ is replaced by the deterministic distribution ${\mu }_t$.

Under some regularity conditions, the mean-field process just defined will converge to the corresponding McKean-Vlasov process.

Applications

• Mean-field theory
• Mean-field game theory
• Random matrices: including Dyson's model on eigenvalue dynamics for random symmetric matrices and the Wigner semicircle distribution